When m is divided by 7, the remainder is 2. What is the remainder when 4m is divided by 7?
Please give me an explanation and not just the answer! Thankss:)
Well, well, well, dividing numbers, are we? Alright, let's have some mathematical fun!
When we say that "m is divided by 7, the remainder is 2," we can write it as an equation: m ≡ 2 (mod 7).
Here's the twist! We want to find the remainder when 4m is divided by 7, so let's multiply both sides of the equation by 4 (because that's the magical number!).
4m ≡ 8 (mod 7)
Now, we can simplify the left side. Since 4 is congruent to 1 modulo 7, we can rewrite it as:
m ≡ 2 (mod 7)
4m ≡ 2 (mod 7)
And voila! The remainder when 4m is divided by 7 is also 2. Easy as pie, right?
But hey, if you're still not convinced, try dividing some numbers by 7 and see if they make you laugh or cry! Just remember, remainders can be a real joker sometimes!
To find the remainder when 4m is divided by 7, we need to understand the relationship between m and 4m.
Given that when m is divided by 7, the remainder is 2, we can write this as:
m = 7a + 2, where a is an integer.
Multiplying both sides of this equation by 4, we get:
4m = 4(7a + 2), which simplifies to:
4m = 28a + 8.
Now, let's divide 4m by 7:
(28a + 8) ÷ 7.
The remainder when dividing by 7 is the same as the remainder when dividing the individual terms by 7. So, we have:
28a ÷ 7 = 4a (No remainder)
8 ÷ 7 = 1 remainder 1.
Therefore, the remainder when 4m is divided by 7 is 1.
In summary, when m is divided by 7 and the remainder is 2, multiplying m by 4 will result in a remainder of 1 when divided by 7.
To find the remainder when 4m is divided by 7, we need to analyze the relationship between m and 4m.
Given that when m is divided by 7, the remainder is 2, we can write the equation as follows:
m ≡ 2 (mod 7)
This means that m is congruent to 2 modulo 7, which can be interpreted as m being of the form:
m = 7k + 2
where k is an integer. This equation implies that m is 2 more than some multiple of 7.
Now, let's find the remainder when 4m is divided by 7. We can express 4m as follows:
4m = 4(7k + 2)
Distributing the 4:
4m = 28k + 8
Now, let's consider the expression 28k + 8 in terms of modulo 7. We can rewrite it as:
28k + 8 ≡ (28k + 7) + 1 ≡ 7(4k + 1) + 1
Since 7 is a multiple of 7, it leaves no remainder when divided by 7. Therefore, we can simplify the equation to:
28k + 8 ≡ 1 (mod 7)
This means that 28k + 8 is congruent to 1 modulo 7, which implies:
4m ≡ 1 (mod 7)
Hence, when 4m is divided by 7, the remainder is 1.
To summarize, when m is divided by 7 with a remainder of 2, multiplying m by 4 and dividing the result by 7 will yield a remainder of 1.
When 4m is divided by 7, you would have 4*2=8 left over, BUT since you are dividing by 7, seven goes into 4m an extra time, leaving 1 left over as the remainder.
Example:
Suppose m = 16
then m/7 = 2 2/7
(That is 2 with 2 left over)
4m = 64 4m/7 = 9 1/7
(That is 9 with 1 left over)
m/7=2 --> m=14
4m/7=x --> 4*14/7=8